Modern petroleum drilling and production operations demand a great quantity of information relating to the parameters and conditions downhole. Such information typically includes the location and orientation of the borehole and drilling assembly, earth formation properties, and parameters of the downhole drilling environment. The collection of information relating to formation properties and downhole conditions is commonly referred to as “logging.” Logging can be performed during the drilling process itself (hence the term “logging while drilling” or “LWD,” which is frequently used interchangeably with the term “measurement while drilling” or “MWD”), or at various times during the drilling process with the drillstring removed using a wireline logging tool.
Various measurement tools exist for use in logging. One such tool is the resistivity tool, which includes one or more antennas for transmitting an electromagnetic signal into the formation and one or more antennas for receiving a formation response. When operated at low frequencies, the resistivity tool may be called an “induction” tool, and at high frequencies it may be called an electromagnetic wave propagation tool. Though the physical phenomena that dominate the measurement may vary with frequency, the operating principles for the tool are consistent. In some cases, the amplitude and/or the phase of the receive signals are compared to the amplitude and/or phase of the transmit signals to measure the formation resistivity. In other cases, the amplitude and/or phase of multiple receive signals are compared to each other to measure the formation resistivity.
When plotted as a function of depth or tool position in the borehole, the logging tool resistivity measurements are termed “resistivity logs.” Such logs may provide indications of hydrocarbon concentrations and other information useful to drillers and completion engineers. The resistivity measurements presented in the logs are a complex function of formation anisotropy and dip angle, as well as the logging tool's azimuthal orientation. A number of existing methods based on multi-component induction tool measurements (e.g., triaxial induction tools) may be used to estimate resistivity in transverse-isotropic formations without cross-beddings. However, such methods produce questionable results when applied to cross-bedded formations. Cross-bedding occurs when material is deposited on an inclined surface within a depositional environment that contained a flowing medium such as water or wind, producing groups of inclined formation layers. Because the dip angle of such cross-bedded formations can vary significantly from group to group, the estimation of cross-bedded formation resistivity can be difficult. While methods do exist for estimating formation resistivity within cross-bedded formations, such methods are typically radial 1-dimensional or homogeneous techniques that do not account for layer or formation boundaries and produce significant errors from the effects of shoulder beds. See, e.g., Yin, Hezhu, U.S. Pat. No. 8,360,146 and Wang, Tsili et al., U.S. Pat. No. 7,317,991. Still another method proposes the use of 3D forward modeling in a single-stage inversion that solves for all unknown parameters at once. See Wang, Hanming et al., Sensitivity Study and Inversion of the Fully-Triaxial Induction Logging in Cross-Bedded Anisotropic Formation, SEG Technical Program Expanded Abstracts, 284 (2008). However, this method requires large computation times and processing resources, making such an approach impractical for use in real-time applications.
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